Given that $w = (1/4)\sum_{a,b}{(-1)^{(a.b)}\langle \phi^{+} |(m_a. \sigma)\otimes(n_b. \sigma)|\phi^{+}\rangle }$
where $m_a = (\sin \theta_a, 0, cos \theta_a),\quad n_b = (\sin \theta_b, 0, \cos \theta_b),$
and that $$(Q ⊗ S + R ⊗ S + R ⊗ T − Q ⊗ T)^2 = 4I + [Q, R] ⊗ [S, T],$$ where $[A, B] = AB − BA,$
and that for any state $|\psi\rangle $ and any Hermitian operator $A$, $(\langle ψ|A|ψ\rangle )^2 ≤ \space (\langle ψ|A^2|ψ\rangle ).$
How do I show that $w \leqslant 1/\sqrt{2}\;$?
I attempted to square $w$ and then find an upper bound for it, by expanding out the sum (since $a,b = 0,1$) and I got $w^2 \leqslant (1/4)$